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Point Group Tables of Th(m-3)

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Character Table of the group Th(m-3) *
Th(m-3)#13+3-2-1-3+-3-mfunctions
Mult.-14431443
AgΓ1+11111111x2+y2+z2
1Eg
2Eg
Γ2+
Γ3+
1
1
w
w2
w2
w
1
1
1
1
w
w2
w2
w
1
1
(2z2-x2-y2,x2-y2)
TgΓ4+300-1300-1(xz,yz,xy),(Jx,Jy,Jz)
AuΓ1-1111-1-1-1-1
1Eu
2Eu
Γ2-
Γ3-
1
1
w
w2
w2
w
1
1
-1
-1
-w
-w2
-w2
-w
-1
-1
TuΓ4-300-1-3001(x,y,z)

w = exp(2iπ/3)



Subgroups of the group Th(m-3)
SubgroupOrderIndex
Th(m-3)241
T(23)122
C3i(-3)64
C3(3)38
D2h(mmm)83
C2v(mm2)46
D2(222)46
C2h(2/m)46
C2(2)212
Cs(m)212
Ci(-1)212
C1(1)124

[ Subduction tables ]

Multiplication Table of irreducible representations of the group Th(m-3)
Th(m-3)AgAu1Eg1Eu2Eg2EuTuTg
AgAgAu1Eg1Eu2Eg2EuTuTg
AuAg1Eu1Eg2Eu2EgTgTu
1Eg2Eg2EuAgAuTuTg
1Eu2EgAuAgTgTu
2Eg1Eg1EuTuTg
2Eu1EgTgTu
TuAg+1Eg+2Eg+2TgAu+1Eu+2Eu+2Tu
TgAg+1Eg+2Eg+2Tg

[ Note: the table is symmetric ]


Symmetrized Products of Irreps
Th(m-3)AgAu1Eg1Eu2Eg2EuTuTg
[Ag x Ag]1·······
[Au x Au]1·······
[1Eg x 1Eg]····1···
[1Eu x 1Eu]····1···
[2Eg x 2Eg]··1·····
[2Eu x 2Eu]··1·····
[Tu x Tu]1·1·1··1
[Tg x Tg]1·1·1··1


Antisymmetrized Products of Irreps
Th(m-3)AgAu1Eg1Eu2Eg2EuTuTg
{Ag x Ag}········
{Au x Au}········
{1Eg x 1Eg}········
{1Eu x 1Eu}········
{2Eg x 2Eg}········
{2Eu x 2Eu}········
{Tu x Tu}·······1
{Tg x Tg}·······1


Irreps Decompositions
Th(m-3)AgAu1Eg1Eu2Eg2EuTuTg
V······1·
[V2]1·1·1··1
[V3]·1····3·
[V4]2·2·2··3
A·······1
[A2]1·1·1··1
[A3]1······3
[A4]2·2·2··3
[V2]xV·1·1·15·
[[V2]2]3·3·3··4
{V2}·······1
{A2}·······1
{[V2]2}1·1·1··4

V ≡ the vector representation
A ≡ the axial representation


IR Selection Rules
IRAgAu1Eg1Eu2Eg2EuTuTg
Agx
Aux
1Egx
1Eux
2Egx
2Eux
Tuxxxx
Tgxxxx

[ Note: x means allowed ]


Raman Selection Rules
RamanAgAu1Eg1Eu2Eg2EuTuTg
Agxxxx
Auxxxx
1Egxxxx
1Euxxxx
2Egxxxx
2Euxxxx
Tuxxxx
Tgxxxx

[ Note: x means allowed ]


Irreps Dimensions Irreps of the point group
Subduction of the rotation group D(L) to irreps of the group Th(m-3)
L2L+1AgAu1Eg1Eu2Eg2EuTuTg
011·······
13······1·
25··1·1··1
37·1····2·
491·1·1··2
511···1·13·
6132·1·1··3
715·1·1·14·
8171·2·2··4
919·2·1·15·
10212·2·2··5



* George F. Koster, John O. Dimmock, Robert G. Wheeler, Hermann Statz (1963). Properties of the thirty-two point groups. Published by the M.I.T. press, Cambridge, Massachusetts.
* Simon L. Altmann and Peter Herzig (1994). Point-Group Theory Tables. Oxford Science Publications.


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